Bayesian Knowledge Tracing: how a tutor knows it's secure

Your child gets three fractions questions right in a row. Have they got fractions, or did they get lucky? A good teacher can tell the difference by watching. A piece of software has to work it out from the answers alone.

Bayesian Knowledge Tracing is how Archimedes does that. It's a decades-old idea from tutoring research, and it's the reason our parent view says "secure" or "nearly there" instead of "45 minutes on the app". Here's how it works, with the actual maths and a worked example you can check yourself.

> Key takeaway: Bayesian Knowledge Tracing keeps one honest estimate per skill. Every answer nudges it, lucky guesses and tired slips are expected, and a skill counts as secure only after the evidence adds up.

What is Bayesian Knowledge Tracing?

Bayesian Knowledge Tracing (BKT) is a model that estimates the chance a student has mastered a skill, based on their right and wrong answers over time. Albert Corbett and John Anderson introduced it in their paper Knowledge tracing: Modeling the acquisition of procedural knowledge, published in User Modeling and User-Adapted Interaction in 1995.

It has had a long life since. A 2024 paper in the same journal was titled Twenty-five years of Bayesian knowledge tracing: a systematic review, and Berkeley researchers maintain pyBKT, an open-source Python library for fitting BKT models.

The core idea is simple. You can't see knowledge directly. You can only see answers. So BKT treats "knows the skill" as a hidden state and updates its belief every time a new answer comes in.

The four numbers behind every skill

Standard BKT describes each skill with four probabilities:

• Prior (p-init): how likely the student already knows the skill before we start.
• Learn (p-transit): how likely they are to learn it from one practice opportunity.
• Slip (p-slip): how likely they are to get it wrong even though they know it. The tired-Tuesday mistake.
• Guess (p-guess): how likely they are to get it right without knowing it. The lucky guess.

Slip and guess are what make BKT kind. One wrong answer doesn't wipe out a skill, and one right answer doesn't prove it.

The update, in two steps

After each answer, BKT does two things.

Step 1: weigh the evidence. Call the current chance the skill is known K, slip S and guess G. If the answer was right, the new belief is:

K × (1 − S)  ÷  [ K × (1 − S) + (1 − K) × G ]

If it was wrong:

K × S  ÷  [ K × S + (1 − K) × (1 − G) ]

Step 2: allow for learning. Practice itself teaches, so the estimate moves up a little more. Taking the belief from step 1 as B:

next K  =  B + (1 − B) × T

Here T is the learn rate. The standard model assumes no forgetting, so a known skill stays known. That's one of the simplifications later variants relax.

How Bayesian Knowledge Tracing works: a worked example

Let's follow one child through one skill, say adding fractions with the same denominator. These are example numbers we picked for illustration, not the values Archimedes uses: prior 20%, learn 15%, slip 10%, guess 20%.

• Start: 20%
• Right answer: 60%
• Right answer: 89%
• Wrong answer: 58%
• Right answer: 88%
• Right answer: 98%

Notice what happened at the wrong answer. The estimate dropped, but not to zero, because a slip was always possible. Two more right answers later, it was higher than before.

So when does a skill count as mastered? One common research choice is an estimate of 95% or higher. The pyBKT paper from EDM 2021 uses exactly that threshold when it tests how accurately standard BKT estimates mastery. In our example, the child crosses it on the fifth answer.

You can check every number above with the two formulas and a calculator.

Why not just count right answers?

Because a percentage-correct score forgets the order. "Four out of five" looks the same whether the mistake came first, while the child was still learning, or last, after they seemed to have it.

BKT cares about the sequence. Early mistakes matter less once the recent answers are solid, and a sudden run of errors pulls the estimate down quickly.

It also beats time-on-task. Minutes logged tell you a child was present. They don't tell you what they can now do on their own, which is the question our guide to helping kids with homework without doing it for them keeps coming back to.

How does Archimedes use Bayesian Knowledge Tracing?

In Archimedes, BKT sits in the recommendation layer under our tutor model, Archimedes Think. Every skill on the curriculum skill map carries its own estimate, and every answer nudges it. That estimate helps choose the next question: build on what's secure, or step back to an earlier idea that's still shaky.

It also powers what parents see. The parent view shows what's secure, what's shaky and where your child is stuck, skill by skill, based on their answers over time rather than how long they were logged in.

The tutor's job is to get honest evidence in the first place. As our algorithm page explains, each turn is short, asks at most one question, and doesn't hand over the answer on the first try. That design matters for the maths too. An answer the AI gave away tells BKT nothing about the child, which is the same "illusion of knowledge" problem we covered in AI in schools: what AERO's new advice means for teachers.

Where BKT falls short

We'd rather be upfront about the limits:

• Standard BKT doesn't model forgetting. A skill marked secure in March can fade by August. Variants exist that handle this.
• It treats each skill separately. Real skills overlap, which is why the skill map and its prerequisites matter as much as the estimate.
• The four numbers have to come from somewhere. Guess and slip rates differ by skill and by question, and getting them right takes real data.

FAQ

What is Bayesian Knowledge Tracing in simple terms?

Bayesian Knowledge Tracing is a way for a tutoring system to estimate how likely a student is to have mastered a skill. It starts with a guess, then updates that estimate after every right or wrong answer, allowing for lucky guesses and careless slips along the way.

Who invented Bayesian Knowledge Tracing?

Albert Corbett and John Anderson introduced knowledge tracing in a paper published in 1995 in User Modeling and User-Adapted Interaction. It grew out of their work on computer tutors, and researchers have extended it many times since, including versions that personalise the parameters for each student.

What are the four parameters of BKT?

The four parameters are the prior (chance the skill is already known), the learn rate (chance of learning it from one practice opportunity), slip (chance of a wrong answer despite knowing it) and guess (chance of a right answer without knowing it). Standard BKT sets these separately for each skill.

Is Bayesian Knowledge Tracing the same as deep learning?

No. Standard BKT is a small probabilistic model with four numbers per skill, so you can read and check every estimate. Deep knowledge tracing uses neural networks instead. BKT's big advantage is that its estimates are easy to explain to a parent or teacher.

Want more plain-English notes on how AI tutors actually work? Get the next issue by email. We also write up the build itself, like how we set up this edtech newsletter.

By Thomas Heal, founder of Archimedes. Thomas is building a patient AI tutor for parents, students and teachers from Perth, Western Australia.